In a triangle ABC if `b=2,B=30^@` then the area of the circumcircle of triangle ABC in square units is :

Options

A. `pi`
B. `2pi`
C. `4pi`
D. `6pi`

Answer

To find the area of the circumcircle of triangle ABC given that \( b = 2 \) and \( B = 30^\circ \), we can follow these steps:

### Step 1: Use the Sine Rule
According to the Sine Rule, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
where \( R \) is the circumradius of the triangle.

### Step 2: Substitute Known Values
We know:
- \( b = 2 \)
- \( B = 30^\circ \)

From the Sine Rule, we can express \( R \) in terms of \( b \) and \( B \):
\[ \frac{b}{\sin B} = 2R \]
Substituting the known values:
\[ \frac{2}{\sin 30^\circ} = 2R \]

### Step 3: Calculate \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]

### Step 4: Substitute \( \sin 30^\circ \) into the Equation
Now substituting \( \sin 30^\circ \) into the equation:
\[ \frac{2}{\frac{1}{2}} = 2R \]
This simplifies to:
\[ 2 \times 2 = 2R \implies 4 = 2R \]

### Step 5: Solve for \( R \)
Dividing both sides by 2:
\[ R = 2 \]

### Step 6: Calculate the Area of the Circumcircle
The area \( A \) of the circumcircle is given by the formula:
\[ A = \pi R^2 \]
Substituting the value of \( R \):
\[ A = \pi (2)^2 = \pi \times 4 = 4\pi \]

### Final Answer
Thus, the area of the circumcircle of triangle ABC is:
\[ \boxed{4\pi} \text{ square units} \]

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